A systematic approach to the modelling and comparison of the geometries of spherical electrodes in inertial electrostatic confinement fusion devices

Inertial electrostatic confinement fusion (IECF) devices often use two concentric spherical electrodes to converge ions in a plasma electrostatically. Using a highly transparent inner cathode, the ions can move through the cathode and collide at the center to undergo fusion reactions. This is a simple method to build a neutron source. Past research has focused chiefly on cathode “grids” manufactured by joining metal wire loops or disc-shaped elements via spot welding. There are two common geometries: “Globe” grids with a distinct latitude-longitude structure and “symmetric” grids with even-sized triangular-shaped apertures. Recent advances in additive manufacturing have opened the way to manufacturing a third class of grids in which the apertures are evenly distributed over the grid surface and have either circular or polygonal shaped apertures - here called “regular” grids. These three types are analyzed and compared based on a set of metrics, including transparency, homogeneity of aperture size, and the regularity of aperture distribution. It is shown that every type of grid comes with different advantages and disadvantages. The analysis focuses on grid geometries with 6 to 120 apertures.


A.1 Calculation of Geometry
For the calculation of the grid coordinates it is beneficial to use spherical coordinates wherever possible.Each aperture can be associated with a constant polar angle ∆θ and a constant azimuthal angle ∆ϕ, which can be calculated from the numbers of longitudinal segments n long and latitudinal segments n lat : As it can be seen from Fig.
A.1, the geometry of the apertures close to the poles diverges from the other ones.Real cathode grids in IEC devices might feature a slightly different alignment of the longitudinal segments at the top and bottom.In order to not lose the generality of the approach presented here, it is assumed that all longitudinal elements cross at the top and bottom.The variable ξ will be used to enumerate the apertures from top to bottom with ξ = 1 . . .(n lat + 1) (see For the description of the aperture coordinates, two sets of points vglobe,up and vglobe,low will be used (see Fig. A.1d).They describe one-half of the corner points of the apertures which are assumed to be symmetric to the x-z plane.The apertures closest to the poles are triangles.The corner points are defined by the intersection of the longitudinal elements (see also the detailed top-view in Fig. A.1c with vglobe,up (ξ , ν = 1)).Therefore, the upmost corner point has the following coordinates. vglobe,up For the calculation of the trapezoidal coordinates the three auxiliary variables x ′ up , y ′ up and z ′ up are defined first: These can now be used to calculate the remaining upper aperture coordinates: Analogously, the lower coordinates can be calculated.Only two of the auxiliary variables need to be adapted: This leads to: For the aperture closest to the south pole the following coordinates apply:

A.2 Calculation of Transparency
The calculation of the transparency is as follows: From the surface area of a spherical zone, it can be derived that an individual aperture, which is in an azimuthal direction essentially described by the azimuthal angle ∆θ (additionally reduced by the bridge angle α bridge ) and in polar direction by the z-coordinates of pglobe,up and pglobe,low (which both directly depend on the polar angle ∆θ and the influence of the bridge angle α bridge ).The individual surface area of one aperture can therefore be described by: A ap,globe,ind (ξ ) = 2R grid z up (ξ ) This equation can be simply solved numerically (e.g. with Matlab 2021b's integral-function).To obtain the transparency of the globe grid, the surface area of all apertures needs to be computed and divided by the surface area of the sphere that has the same diameter as the grid:

A.3 Calculation of Potential Energy
The potential energy of the globe grid is computed from the center-points of which each is associated with an individual aperture by the half of the polar angle θ .Although this is different from the centroid defined by four corner points and also different from the centroid of the surface area, these coordinates describe the position of the individual apertures of the globe grid and will therefore be used for the metric.(In the case of the symmetric grids and regular shaped grids with polygonal apertures the centroid defined by the corner points is actually the centroid that defines the center of the aperture).
The position of the center points pglobe,ap of the apertures are described in spherical coordinates by (see Fig.
These points are then used to calculate the potential energy with Eq. 11.

A.4 Calculation of Circular Transparency
The circular transparency of the globe grid η circ,globe is calculated following the general approach described in Chapter 4. A visualization is presented in Fig. A.2.For each aperture, the minimum angle between its centroid (see Eq. A.16) and the bridge element is used to compute an angle for a spherical cap: From the equation for the surface area of a spherical cap follows: Therefore, the circular transparency of the globe grid η globe,circ becomes: The normalized circular transparency of the globe grids ηglobe,circ is calculated analogously to Eq. 10.

B Regular Grids -Detailed Parameter Calculation B.1 Mathematical Problems Related to Distribution Problem
Before the iterative optimization procedure to achieve a near-optimum aperture distribution is described, a brief overview of related mathematical problems is presented in the following list: • Tammes Problem: Motivated from the distribution of pores on pollen grains, this problem asks for the optimum distribution of equally sized circles on a sphere such that their radius becomes maximized without overlapping 1 .In the present case, the circles represent the base of the spherical caps which define the apertures.• Thomson Problem: To describe the electron distribution in the superseded "plum pudding model" of the atom, Thomson looked for the distribution of N electrons on the surface of a unit sphere in which the overall electrostatic potential energy (based on the Coulomb repulsion between the electrons, see Eq. 11) becomes minimal 2 .In the context of this study, the position of the electrons describes the center of the apertures.• Spherical Codes: This is somewhat of an inversion to the two problems above and describes a set of N points on an Euclidean sphere with a defined minimum Euclidean distance.Whereas spherical codes are based on a predefined minimum Euclidean distance, the present problem in this paper requires the identification of a configuration such that the Euclidean distance between a set of points is maximized 3 .

B.2 Aperture Distribution Optimization
The optimization of the distribution follows a three-step process as described by Gautam and Vaintrob 4 .1. Initially, the set of points pi which describes the centroids of the apertures is randomly distributed over the surface of the sphere.2.Then, the points are iteratively redistributed by the gradient of their potential energy.3. Finally, the points are iteratively optimized by the individual maximization of the angles between the two closest neighbors.Similar approaches are also described in 5,6 .
For the initial random distribution of the points pi over the surface of the sphere, it was found to be useful to define the position of two apertures, e.g. two being on the z-axis ("north" and "south" pole).p1 = (0, 0, 1), pN/2+1 = (0, 0, −1) (B.20)

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It can be advantageous to align one or more additional points with a reference such as the x-z plane.In the second step the gradient of the potential energy E pot defined in Eq. 11 is used to redistribute the points: The points pi are then redistributed by: The factor ε 1 weighs the gradient and Gautam 4 suggests ε 1 ≈ 10 −3 .The next step of the optimization process involves the identification of the closest neighbor p j for each point pi and both points are relocated in opposite directions.The closest neighbor for each point is identified by the dot product: The subsequent optimization is then achieved by moving each pair of closest points -here denoted as p1 and p1 into the opposite direction with the weighting function ε 2 : If necessary, the antipodal requirement has to be taken into account by realigning the distribution by Eq. 4. Gautam 4 suggests the following function which decreases with the number of optimization steps n dot :

B.3.1 Circular Apertures
The circular apertures are described by conical cuts into the unit sphere.The surface areas of these apertures are that of spherical caps, which are defined by the half-angle θ .As stated in Section 2.2, it is recommended that all the apertures should have the same size.Therefore, it is necessary to find the minimum angle between the closest apertures by using Eq.B.23.The minimum half angle θ min is one half of this angle (see also Fig. 7).
With this the maximum theoretical transparency for a grid with spherical apertures and a bridge thickness t bridge = 0 can be calculated.Under these conditions the area of the spherical cap A cap , which defines a single aperture, becomes: The maximum theoretical grid transparency η circular,max with equal sized apertures of half angle θ min is then defined as: If a finite bridge thickness t bridge is assumed at the point of the smallest distance between two apertures, the reduced surface area of the caps A cap, red based on the half-angle α bridge (see Eq. 5) becomes: The transparency η circular is then calculated analogously to Eq. B.29:

B.3.2 Polygonal Apertures
The calculation of the transparency for polygonal apertures is less trivial compared with the spherical apertures.Assuming a constant size of the structures between the apertures, the structures can be described by so-called spherical Voronoi diagrams.
In the following paragraph, first, the basics of spherical Voronoi diagrams are reviewed and how the concept applies to regular-shaped grid geometries.
In its simplest 2D form a Voronoi diagram describes the partition of a plane into regions based on points (usually called sites) located on the plane in each region has the shortest Euclidean distance to one site.In the case of the spherical Voronoi diagram the partition does not happen on a plane but on the surface of a sphere and the sites are defined by the center points of the equally distributed grid apertures.For an in-depth description and the complex calculation process of spherical Voronoi diagrams, see [7][8][9][10][11] .The spherical Voronoi diagrams were computed with the scipy.spatial.SphericalVoronoi class of the Python SciPy library 12 .In the following a brief description of spherical Voronoi diagrams is presented and how individual elements are related to the definition of regular grids with polygonal apertures.
The Voronoi diagrams are constructed from three main elements, which are displayed in Fig. B.3: 1. Bisectors, which form the base for 2. Voronoi edges (which represent the center line of the bridge segments of the regular grids with polygonal apertures) and 3. Voronoi vertices (which represent the joints of the polygonal grids).The following paragraph mainly follows the description of 13 .The bisector B i j is comprised of the set of points that are equidistant to the sites pi and p j : Therefore, the bisector describes a great circle.To describe the Voronoi edge E i j between two adjacent Voronoi cells with the boundaries ∂V ( pi ) and ∂V ( p j ) of their respective convex regions the intersection has to be calculated: The vertices vi, j....k shared by multiple apertures can be described as: In the context of this analysis a Voronoi edge describes the center-arc of a bridge segment and a Voronoi vertex represents the center-point where multiple segments meet.The calculation of the grid transparency is more complicated for grids with polygonal apertures compared to grids with circular apertures.The surface area of an individual spherical Voronoi cell can be calculated by dividing the cell into spherical triangles.One way to calculate the area of a spherical triangle A triangle is to use Girard's theorem: with α, β , γ being the angles of the spherical triangle.However, a less well-known formula presented by Eriksson 14 was found to be more suitable: where the area A triangle is formed by the vectors a, b and c which represent the vector of one site and two corresponding Voronoi vertices that form (one part) of the spherical triangle of a Voronoi cell.To account for the finite thickness of the bridges between the apertures, a set of "reduced" vertices vi,red needs to be created (three vi,red per vi ).The vertices are rotated by the angle α bridge around the normal vector between the vertex vi and the corresponding center point pi .This rotation can be calculated with the Euler-Rodrigues rotation formula 15 .By defining the rotation vector k rot : The reduced vertices vi,red can then be calculated by: vi For quadrilateral, pentagonal and hexagonal-shaped apertures and apertures with more edges, the n i -sided surface areas can be calculated by dividing them into n i spherical triangles (the three vertices defining the triangle are the center-point pi and the two vertices of one of the edges of the aperture), calculate the individual surface areas A triangle,h and then simply compute their total sum A polygonal,i : The transparency of the grid with polygonal apertures η polygonal is calculated by dividing the total sum of the individual areas of the polygonal apertures A polygonal,i by the total area of the underlying sphere.
An important question is how many edges the grid apertures will have in order to estimate how regularly the grids will be shaped.Euler's theorem for convex polyhedra with f faces, e edges and v vertices states 16 : This equation shows that the average number of apertures is less than 6.In the case of 120 apertures, the average number of edges per aperture becomes 5.9.For example, the equation predicts correctly the average number of edges per aperture for the Buckyball geometry (5.625 for 20 hexagons and 12 pentagons).

B.4 Implementation of Code
The code was implemented into Python 3. By making extensive use of the NumPy-library 17 (especially the aggregations and broadcasting capability to speed up the calculation) and the SciPy's scipy.spatial.SphericalVoronoi class 12 the code was kept simple.
The following parameters were used for the optimization study:

bridge sin ∆ϕ 2 (
Fig. A.1a and analogously as shown in Fig. A.1b ν = 1 . . .2n long in the azimuthal direction).The top region (as displayed in Fig. A.1c) can be described by the top radius r top and polar top angle θ top , which are defined by: r top = R grid sin α Polar angle ∆θ (side view).Azimuthal angle ∆ϕ (top view). bridge  top y x  globe ,up (,  = ) (c) Intersection of longitudinal elements at "north" pole (detailed top view).Aperture Coordinates (side view, t bridge increased for visual clarity).

Figure A. 1 .
Figure A.1.Geometric properties of globe grid with conformal cross-section and equiangular distribution of latitudinal segments by constant polar angle ∆θ (see (a)) and equiangular distribution of longitudinal segments by constant azimuthal angle ∆ϕ (see (b)).(c) shows detailed view of the intersection at the top section, which is described by the top radius r top and top angle θ top .(d) describes the centerpoints of apertures pglobe,ap as well as the vertices of the cornerpoints of the apertures vglobe,up and vglobe,down .

Figure A. 2 .
Figure A.2. Illustration of circular transparency η circ for globe grid configurations.The outline of the spherical cap with base radius r circ (with half-angle (θ min − α bridge )) and its centerpoint pglobe,ap (ξ = 3, ν = 1) is highlighted by the blue ring.

Figure B. 3 .
Figure B.3.Spherical Voronoi structure with sites pi , p j and pk superimposed onto regular shaped grid structure with polygonal shaped apertures based on a dodecahedron (N = 12).

( 6
of faces f is substituted with the equal number of apertures N and it is taken into account that for each n-sided aperture the bridge (edge) is shared by two apertures and every corner (vertices) is shared by three other apertures, the following equation with N n being the number of n-sided apertures is obtained:∑ n (N n n/3 − N n n/2 + N n ) = 2 (B.42)This equation can be simplified to:∑ n − n)N n = 12 (B.43)Therefore, the average number of edges per aperture becomes: